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Multiple choice maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

If $r$ is the radius of the inscribed circle of a regular polygon of $n$ sides, then $r$ is equal to?

  1. $\dfrac{a}{2} cot (\dfrac{\pi}{2n})$
  2. $\dfrac{a}{2} cot (\dfrac{\pi}{n})$
  3. $\dfrac{a}{2} tan (\dfrac{\pi}{n})$
  4. $\dfrac{a}{2} cos (\dfrac{\pi}{n})$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

in $\Delta{ABL}$, $AL$ is the radius of the inscribed circle. 


$ BL=\cfrac{BC}{2}=\cfrac{a}{2}$

$\cot(\cfrac{\pi}{n})=\cfrac{AL}{BL}=\cfrac{r}{\dfrac{a}{2}}$

Hence $r=\cfrac{a(\cot(\dfrac{\pi}{n}))}{2}$

Multiple choice maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

The area of a regular polygon of $2n$ sides inscribed in a circle is given by?

  1. The geometric mean of the areas of the inscribed and circumscribed polygons of $n$ sides.
  2. The arithmetic mean of the areas of the inscribed and circumscribed polygons of $n$ sides.
  3. The harmonic mean of the areas of the inscribed and circumscribed polygons of $n$ sides.

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let $a$ be the radius of the circle 


Then,$\displaystyle s _{1}= $ Area of regular polygon of n sides inscribed in the circle $\displaystyle =\frac{1}{2}na^{2}\sin\left ( \frac{2\pi }{n} \right )$

$\displaystyle s _{2}= $  Area of regular polygon of n sides circumscribing in the circle $\displaystyle  = na^{2}\tan \frac{\pi }{n}$

$\displaystyle s _{3}= $ Area of regular polygon of 2n sides inscribed in the circle $\displaystyle  = na^{2}\tan \frac{\pi }{n}$ 

[replacing $n$ by $2n$ is $\displaystyle {(S _{1}}$]

$\displaystyle \therefore $ Geometric mean of $\displaystyle {S _{1}}$ and 

$\displaystyle {S _{2}}$ $\displaystyle = \sqrt{(S _{1}S _{2})}= na^{2}\sin\left ( \frac{\pi }{n}\right ) = S _{3}$

Multiple choice maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

If A B C D E F is a regular hexagon with A B = a and B C = b, then CE equals

  1. b-a

  2. -b

  3. b-2a

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In a regular hexagon ABCDEF, the vector CE can be found using vector addition. Since AB = a and BC = b, the vectors for the sides are related by the geometry of the hexagon, leading to CE = b - a.

Multiple choice maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

If  $\alpha$  is the angle which each side of a regular polygon of  $n$  sides subtends at its centre, then  $1 + \cos \alpha + \cos 2 \alpha + \cos 3 \alpha \ldots + \cos ( n - 1 ) \alpha$  is equal to

  1. $n$
  2. $0$
  3. $1$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For a regular polygon of n sides with central angle alpha = 2pi/n, the sum 1 + cos(alpha) + cos(2alpha) + ... + cos((n-1)alpha) represents the real part of the sum of roots of unity, which geometrically sums to zero because the vectors representing the sides form a closed polygon.